with \(\overleftarrow{\delta }\)  signifying that the derivative acts to the left.
The \(\cdot \)  implies a continuous matrix multiplication including sum over all field components and integration over spacetime. The \( {\rm STr } \)  denotes a supertrace in the same sense with a minus sign inserted for anti-commuting fields.
For gravity the RG kernel now has component for each field \(\Psi _k = \lbrace  \Psi _{\mu \nu }^g, \Psi ^{c\mu }, \Psi ^{\bar{c}}_\mu \rbrace \) , such that
\(\Psi _k = \langle \partial _t \hat{f}_k \rangle _{f,k}\)  .
By setting \(\Psi _k =0\)  we obtain the flow equation for gravity derived in [1]}, however in this case we would have to also compute the flow of inessential couplings.
Using the background field method [2]}, one is ultimately interested in identifying \(\bar{g}_{\mu \nu } = g_{\mu \nu }\)  and setting \(c^\mu =0 = \bar{c}_\nu \) . It is, therefore, convenient to write the action as
\(\Gamma _k[g, c ,\bar{c};\bar{g}]= \bar{\Gamma }_k[g] + \hat{\Gamma }_k[g, c ,\bar{c};\bar{g}] \,,\) 
